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Locally optimized RANSAC

Locally optimized RANSAC
Locally optimized RANSAC

Locally Optimized RANSAC?

Ondˇr ej Chum1,2,Jiˇr′?Matas1,2,and Josef Kittler2

1Center for Machine Perception,Czech Technical University,Faculty of Electrical Engineering Dept.of Cybernetics,Karlovo n′a m.13,12135Prague,Czech Republic

2CVSSP,University of Surrey,Guildford GU27XH,United Kingdom

Abstract

A new enhancement of RANSAC,the locally optimized RANSAC(LO-RANSAC),is intro-

duced.It has been observed that,to?nd an optimal solution(with a given probability),

the number of samples drawn in RANSAC is signi?cantly higher than predicted from the

mathematical model.This is due to the incorrect assumption,that a model with parame-

ters computed from an outlier-free sample is consistent with all inliers.The assumption

rarely holds in practice.The locally optimized RANSAC makes no new assumptions

about the data,on the contrary-it makes the above-mentioned assumption valid by

applying local optimization to the solution estimated from the random sample.

The performance of the improved RANSAC is evaluated in a number of epipolar

geometry and homography estimation https://www.wendangku.net/doc/ac5036638.html,pared with standard RANSAC,

the speed-up achieved is two to three fold and the quality of the solution(measured by

the number of inliers)is increased by10-20%.The number of samples drawn is in good

agreement with theoretical predictions.

1Introduction

Many computer vision algorithms include a robust estimation step where model param-

eters are computed from a data set containing a signi?cant proportion of outliers.The RANSAC algorithm introduced by Fishler and Bolles in1981[3]is possibly the most widely used robust estimator in the?eld of computer vision.RANSAC has been applied

in the context of short baseline stereo[13,12],wide baseline stereo matching[9,15,

10,6],motion segmentation[13],mosaicing[7],detection of geometric primitives[2],

robust eigenimage matching[5]and elsewhere.

In a classical formulation of RANSAC,the problem is to?nd all inliers in a set of

data points.The number of inliers I is typically not known a priori.Inliers are data

points consistent with the’best’model,e.g.epipolar geometry or homography in a two

view correspondence problem or line or ellipse parameters in the case of detection of

geometric primitives.The RANSAC procedure?nds,with a certain probability,all inliers

and the corresponding model by repeatedly drawing random samples from the input set

of data points.

?The authors were supported by the European Union under projects IST-2001-32184,

ICA1-CT-2000-70002and by the Czech Ministry of Education under project LN00B096and

by the Czech Technical University under project CTU0306013.The images for experiments

A,B,and E were kindly provided by T.Tuytelaars(VISICS,K.U.Leuven),C by M.Pollefeys

(VISICS,K.U.Leuven),and E by K.Mikolajczyk(INRIA Rh?o ne-Alpes).

RANSAC is popular because it is simple and it works well in practice.The reason is that almost no assumptions are made about the data and no(unrealistic)conditions have to be satis?ed for RANSAC to succeed.However,it has been observed experimentally that RANSAC runs much longer(even by an order of magnitude)than theoretically pre-dicted[11].The discrepancy is due to one assumption of RANSAC that is rarely true in practice:it is assumed that a model with parameters computed from an uncontaminated sample is consistent with all inliers.

In this paper we propose a novel improvement of RANSAC exploiting the fact that the model hypothesis from an uncontaminated minimal sample is almost always suf-?ciently near the optimal solution and a local optimization step applied to selected models produces an algorithm with near perfect agreement with theoretical(i.e.opti-mal)performance.This approach not only increases the number of inliers found and consequently speeds up the RANSAC procedure by allowing its earlier termination,but also returns models of higher quality.The increase of average time spent in a single RANSAC veri?cation step is minimal.The proposed optimization strategy guarantees that the number of samples to which the optimization is applied is insigni?cant.

The main contributions of this paper are(a)modi?cation of the RANSAC that si-multaneously improve the speed of the algorithm and and the quality of the solution (which is near to optimal)(b)introduction of two local optimization methods(c)a rule for application of the local optimization and a theoretical analysis showing the local optimization is applied at most log k times,where k is the number of samples drawn.In experiments on two image geometry estimation(epipolar geometry and homography) the speed-up achieved is two to three fold.

The improvement proposed in this paper requires no extra input information or prior knowledge,and it does not interfere with other modi?cations of the algorithm,the MLE-SAC[14],R-RANSAC[1]and NAPSAC[8].MLESAC,proposed by Torr and Zisserman, de?nes a cost function in the maximal likelihood framework.

The structure of this paper is as follows.First,in Section2,the motivation of this paper is discussed in detail and the general algorithm of locally optimized RANSAC is described.Four different methods of local optimization are proposed in Section3. All methods are experimentally tested and evaluated through epipolar geometry and homography estimation.The results are shown and discussed in Section4.The paper is concluded in Section5.

2Algorithm

The structure of the RANSAC algorithm is simple but powerful.Repeatedly,subsets are randomly selected from the input data and model parameters?tting the sample are computed.The size of the random samples is the smallest suf?cient for determining model parameters.In a second step,the quality of the model parameters is evaluated on the full data set.Different cost functions may be used[14]for the evaluation,the standard being the number of inliers,i.e.the number of data points consistent with the model.The process is terminated[3,13]when the likelihood of?nding a better model becomes low,i.e.the probabilityηof missing a set of inliers of size I within k samples falls under prede?ned threshold

η=(1?P I)k.(1)

Repeat until the probability of?nding better solution falls under prede?ned threshold,as in(1):

1.Select a random sample of the minimum number of data points S m.

2.Estimate the model parameters consistent with this minimal set.

3.Calculate the number of inliers I k,i.e.the data points their error is smaller

than prede?ned thresholdθ.

4.If new maximum has occurred(I k>I j for all j

tion.Store the best model.

Algorithm1:A brief summary of the LO-RANSAC

Symbol P I stands for the probability,that an uncontaminated sample of size m is ran-domly selected from N data points

P I= I

m

m

=

m?1

j=0

I?j

N?j

≈εm,(2)

whereεis the fraction of inliersε=I/N.The number of samples that has to be drawn to ensure givenηis

k=log(η)/log(1?P I).

From equations(1)and(2),it can be seen,that termination criterion based on prob-abilityηexpects that a selection of a single random sample not contaminated by outliers is followed by a discovery of whole set of I inliers.However,this assumption is often not valid since inliers are perturbed by noise.Since RANSAC generates hypotheses from minimal sets,the in?uence of noise is not negligible,and the set of correspondences the size of which is smaller than I is found.The consequence is an increase in the num-ber of samples before the termination of the algorithm.The effect is clearly visible in the histograms of the number of inliers found by standard RANSAC.The?rst column of Figure2shows the histogram for?ve matching experiments.The number of inliers varies by about20-30%.

We propose a modi?cation that increases the number of inliers found near to the optimum I.This is achieved via a local optimization of so-far-the-best samples.For the summary of the locally optimized RANSAC see Algorithm1.The local optimization step is carried out only if a new maximum in the number of inliers from the current sample has occurred,i.e.when standard RANSAC stores its best result.The number of consistent data points with a model from a randomly selected sample can be thought of as a random variable with unknown(or very complicated)density function.This density function is the same for all samples,so the probability that k-th sample will be the best so far is1/k.Then,the average number of reaching the so-far-the-best sample within k samples is

k 11

x

k

1

1

x

dx+1=log k+1.

Note,that this is the upper bound as the number of correspondences is?nite and dis-crete and so the same number of inliers will occur often.This theoretical bound was con?rmed experimentally,the average numbers of local optimization over an execu-tion of(locally optimized)RANSAC can be found in Table3.For more details about experiments see Section4.

Noise level

A

v e r a g e e r r o r

Noise level

S t a n d a r d d e v i a t i o n o f t h e e r r o r

Fig.1.The average error (left)and the standard deviation of the error for samples of 7,8,9,14and all 100points respectively with respect to the noise level.

3Local Optimization Methods

The following methods of local optimization have been tested.The choice is motivated by the two observations that are given later in this section.

1.Standard.The standard implementation of RANSAC without any local optimization.

2.Simple.Take all data points with error smaller than θand use a linear algorithm to hypothesize new model parameters.

3.Iterative.Take all data points with error smaller that K ·θand use linear algorithm to compute new model parameters.Reduce the threshold and iterate until the threshold is θ.

4.Inner RANSAC .A new sampling procedure is executed.Samples are selected only form I k data points consistent with the hypothesised model of k -th step of RANSAC .New models are veri?ed against whole set of data points.As the sampling is running on inlier data,there is no need for the size of sample to be minimal.On the contrary,the size of the sample is selected to minimize the error of the model parameter estimation.In our experiments the size of samples are set to min(I k /2,14)for epipolar geometry (see results in Section 3)and to min(I k /2,12)for the case of homography estimation.The number of repetitions is set to ten in the experiments presented.

5.Inner RANSAC with iteration.This method is similar to the previous one,the dif-ference being that each sample of the inner RANSAC is processed by method 3.The local optimization methods are based on the two following observations.Observation 1:The Size of Sample

The less information (data points)is used to estimate the model parameters in the pres-ence of noise,the less accurate the model is.The reason for RANSAC to draw minimal samples is that every extra point exponentially decreases the probability of selecting an outlier-free sample,which is approximately εm where m is the size of the sample (i.e.the number of data points included in the sample).

It has been shown in [13],that the fundamental matrix estimated from a seven point sample is more precise than the one estimated form eight points using a linear algorithm [4].This is due to the singularity enforcement in the eight point algorithm.However,

the following experiment shows,that this holds only for eight point samples and taking nine or more points gives more stable results than those obtained when the fundamental matrix is computed from seven points only.

Experiment:This experiment shows,how the quality of a hypothesis depends on the number of correspondences used to calculate the fundamental matrix.For seven points, the seven point algorithm was used[13]and for eight and more points the linear algo-rithm[4]was used.The course of experiment was as follows.Noise of different levels was added to the noise-free image points correspondences divided into two sets of hun-dred correspondences.Samples of different sizes were drawn from the?rst set and the average error over the second was computed.This was repeated1000times for each noise level.Results are displayed in Figure1.

This experiment demonstrates,that the more points are used to estimate the model (in this case fundamental matrix)the more precise solution is obtained(with the ex-ception of eight points).The experiment also shows that the minimal sample gives hypotheses of rather poor quality.One can use different cost functions that are more complicated than simply the number of inliers,but evaluating this function only at pa-rameters arising from the minimal sample will get results at best equal to the proposed method of local optimization.

Observation2:Iterative Scheme

It is well known from the robust statistic literature,that pseudo-robust algorithms that ?rst estimate model parameters from all data by least squares minimization,then re-move the data points with the biggest error(or residual)and iteratively repeat this pro-cedure do not lead to correct estimates.It can be easily shown,that a single far–outlying data point,i.e.leverage point,will cause a total destruction of the estimated model pa-rameters.That is because such a leverage point overweights even the majority of inliers in least-squares minimization.This algorithm works only well,when the outliers are not overbearing,so the majority of inliers have bigger in?uence on the least squares.

In local optimization method3there are no leverage points,as each data point has error below K·θsubject to the sampled model.

4Experimental Results

The proposed algorithm was extensively tested on the problem of estimation of the two view relations(epipolar geometry and homography)from image point correspondences. Five experiments are presented in this section,all of them on publicly available data, depicted in Figures3and4.In experiments A and B,the epipolar geometry is estimated in a wide-baseline setting.In experiment C,the epipolar geometry was estimated too, this time from short-baseline stereo images.From the point of view of RANSAC use,the narrow and wide baseline problems differ by the number of correspondences and inliers (see Table1),and also by the distribution of errors of outliers.Experiments D and E try to recover homography.The scene in experiment E is the same as in experiment A and this experiment could be seen as a plane segmentation.All tentative correspondences were detected and matched automatically.

Algorithms were implemented in C and the experiments were ran on AMD K7 1800+MHz processor.The terminating criterion based on equation(1)was set

A B C D E

1

2

3

4

5

Fig.2.Histograms of the number of inliers.The methods1to5(1stands for standard RANSAC)

are stored in rows and different dataset are shown in columns(A to E).On each graph,there is a

number of inliers on the x-axis and how many times this number was reached within one hundred

repetitions on the y-axis.

toη<0.05.The thresholdθwas set toθ=3.84σ2for the epipolar geometry and

θ=5.99σ2for the homography.In both cases the expectedσwas set toσ=0.3.

The characterization of the matching problem,such as number of correspondences,

the total number of inliers and expected number of samples,are summarized in Table1.

The total number of inliers was set to the maximal number of inliers obtained over all

methods over all repetitions.The expected number of samples was calculated according

to the termination criterion mentioned above.

Performance of local optimization methods1to5was evaluated on problems A to

E.The results for100runs are summarized in Table2.For each experiment,a table con-

taining the average number of inliers,average number of samples drawn,average time

spent in RANSAC(in seconds)and ef?ciency(the ratio of the number of samples drawn

and expected)is shown.Table3shows both,how many times the local optimization has

been applied and the theoretical upper bound derived in Section2.

The method5achieved the best results in all experiments in the number of samples

and differs slightly from the theoretically expected number.On the other hand standard RANSAC exceeds this limit2.5–3.3times.In Figure2the histograms of the sizes of the resulting inliers sets are shown.Each column shows results for one method,each

row for one experiment.One can observe that the peaks are shifting to the higher values

with the increasing identi?cation number of method.

Method5reaches the best results in terms of sizes of inlier sets and consequently in

number of samples before termination.This method should be used when the fraction

of inliers is low.Resampling,on the other hand,might be quite costly in the case of high

number of inliers,especially if accompanied by a small number of correspondences in

total)as could be seen in experiment A(61%of inliers out of94correspondences).In

this case,method3was the fastest.Method3obtained signi?cantly better results than

the standard RANSAC in all experiments,the speed up was about100%,and slightly

worse than for method5.We suggest to use method5.Method3might be used in real-

Fig.3.Image pairs and detected points used in epipolar geometry experiments(A-C). Inliers are marked as dots in left images and

outliers as crosses in right images.

Fig.4.Image pairs and detected points used in homography experiments(D and E).In-liers are marked as dots in left images and outliers as crosses in right images.

A B C D E

#corr9494150016094

#inl57274813017

ε61%29%32%19%18%

#sam11534529885228733837 Table 1.Characteristics of experiments A-E.Total number of correspondences,maximal number of inliers found within all tests,fraction of inliersεand theoretically expected number of samples.

12345 inl49.753.955.956.056.2

A sam383205129117115

time0.0180.0100.0070.0100.019

eff 3.35 1.79 1.12 1.02 1.01

inl23.324.425.025.525.7

B sam9081663391499624401639886

time 3.911 2.729 2.154 1.901 1.731

eff 2.63 1.84 1.45 1.27 1.16

inl423.5446.2467.5468.9474.9 C sam252051656411932109479916

time 4.114 2.707 1.971 1.850 1.850

eff 2.85 1.87 1.35 1.24 1.12

inl23.926.728.128.829.0

D sam86525092393635093316

time0.9220.5430.4230.3870.391

eff 3.01 1.77 1.37 1.22 1.15

inl13.514.615.315.715.9

E sam120428551684656135254

time0.9790.6960.5590.4630.444

eff 3.14 2.23 1.78 1.46 1.37 Table2.The summary of local optimization ex-periments:average number of inliers(inl)and samples taken(sam),average time in seconds and ef?ciency(eff).The best values for each row are highlighted in bold.For more details see the description in text in Section4.

time procedures when a high number of inliers is expected.Methods2and4are inferior to methods with iteration(3and5respectively)without any time saving advantage.

5Conclusions

An inprovement of the RANSAC algorithm was introduced.The number of detected inliers increased,and consequently the number of samples drawn decreased.In all ex-periments,the running-time is reduced by a factor of at least two,which may be very

12345

A3.0 5.9 2.6 5.3 2.0 4.9 1.9 4.8 1.8 4.7 B6.411.46.111.15.910.86.010.75.910.6 C7.710.16.89.7 6.59.4 6.79.3 6.59.2

12345

D5.29.14.88.54.58.34.48.24.08.1 E4.89.44.39.14.28.84.08.63.98.6

Table3.The average number of local optimizations ran during one execution of RANSAC and logarithm of average number of samples for comparison.

important in real-time application incorporating a RANSAC step.It has been shown and experimentally veri?ed that the number of local optimization steps is lower than logarithm of the number of samples drawn,and thus local optimization does not slow the procedure down.Four different methods of local optimization were tested and the ef?ciency of method5is almost1.The proposed improvement allows to make pre-cise quantitative statements about the number of samples drawn in RANSAC.The local optimization step applied to selected models produces an algorithm with near perfect agreement with theoretical(i.e.optimal)performance.

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SIFT算法原理

3.1.1尺度空间极值检测 尺度空间理论最早出现于计算机视觉领域,当时其目的是模拟图像数据的多尺度特征。随后Koendetink 利用扩散方程来描述尺度空间滤波过程,并由此证明高斯核是实现尺度变换的唯一变换核。Lindeberg ,Babaud 等人通过不同的推导进一步证明高斯核是唯一的线性核。因此,尺度空间理论的主要思想是利用高斯核对原始图像进行尺度变换,获得图像多尺度下的尺度空间表示序列,对这些序列进行尺度空间特征提取。二维高斯函数定义如下: 222()/221 (,,)2x y G x y e σσπσ-+= (5) 一幅二维图像,在不同尺度下的尺度空间表示可由图像与高斯核卷积得到: (,,(,,)*(,)L x y G x y I x y σσ)= (6) 其中(x,y )为图像点的像素坐标,I(x,y )为图像数据, L 代表了图像的尺度空间。σ称为尺度空间因子,它也是高斯正态分布的方差,其反映了图像被平滑的程度,其值越小表征图像被平滑程度越小,相应尺度越小。大尺度对应于图像的概貌特征,小尺度对应于图像的细节特征。因此,选择合适的尺度因子平滑是建立尺度空间的关键。 在这一步里面,主要是建立高斯金字塔和DOG(Difference of Gaussian)金字塔,然后在DOG 金字塔里面进行极值检测,以初步确定特征点的位置和所在尺度。 (1)建立高斯金字塔 为了得到在不同尺度空间下的稳定特征点,将图像(,)I x y 与不同尺度因子下的高斯核(,,)G x y σ进行卷积操作,构成高斯金字塔。 高斯金字塔有o 阶,一般选择4阶,每一阶有s 层尺度图像,s 一般选择5层。在高斯金字塔的构成中要注意,第1阶的第l 层是放大2倍的原始图像,其目的是为了得到更多的特征点;在同一阶中相邻两层的尺度因子比例系数是k ,则第1阶第2层的尺度因子是k σ,然后其它层以此类推则可;第2阶的第l 层由第一阶的中间层尺度图像进行子抽样获得,其尺度因子是2k σ,然后第2阶的第2层的尺度因子是第1层的k 倍即3 k σ。第3阶的第1层由第2阶的中间层尺度图像进行子抽样获得。其它阶的构成以此类推。 (2)建立DOG 金字塔 DOG 即相邻两尺度空间函数之差,用(,,)D x y σ来表示,如公式(3)所示: (,,)((,,)(,,))*(,)(,,)(,,)D x y G x y k G x y I x y L x y k L x y σσσσσ=-=- (7) DOG 金字塔通过高斯金字塔中相邻尺度空间函数相减即可,如图1所示。在图中,DOG 金字塔的第l 层的尺度因子与高斯金字塔的第l 层是一致的,其它阶也一样。

SIFT算法实现及代码详解

经典算法SIFT实现即代码解释: 以下便是sift源码库编译后的效果图:

为了给有兴趣实现sift算法的朋友提供个参考,特整理此文如下。要了解什么是sift算法,请参考:九、图像特征提取与匹配之SIFT算法。ok,咱们下面,就来利用Rob Hess维护的sift 库来实现sift算法: 首先,请下载Rob Hess维护的sift 库: https://www.wendangku.net/doc/ac5036638.html,/hess/code/sift/ 下载Rob Hess的这个压缩包后,如果直接解压缩,直接编译,那么会出现下面的错误提示: 编译提示:error C1083: Cannot open include file: 'cxcore.h': No such file or directory,找不到这个头文件。 这个错误,是因为你还没有安装opencv,因为:cxcore.h和cv.h是开源的OPEN CV头文件,不是VC++的默认安装文件,所以你还得下载OpenCV并进行安装。然后,可以在OpenCV文件夹下找到你所需要的头文件了。 据网友称,截止2010年4月4日,还没有在VC6.0下成功使用opencv2.0的案例。所以,如果你是VC6.0的用户请下载opencv1.0版本。vs的话,opencv2.0,1.0任意下载。 以下,咱们就以vc6.0为平台举例,下载并安装opencv1.0版本、gsl等。当然,你也可以用vs编译,同样下载opencv(具体版本不受限制)、gsl等。 请按以下步骤操作: 一、下载opencv1.0 https://www.wendangku.net/doc/ac5036638.html,/projects/opencvlibrary/files/opencv-win/1.0/OpenCV_1.0.exe

等额本息和等额本金计算公式

等额本息和等额本金计算公式 等额本金: 本金还款和利息还款: 月还款额=当月本金还款+当月利息式1 其中本金还款是真正偿还贷款的。每月还款之后,贷款的剩余本金就相应减少: 当月剩余本金=上月剩余本金-当月本金还款 直到最后一个月,全部本金偿还完毕。 利息还款是用来偿还剩余本金在本月所产生的利息的。每月还款中必须将本月本金所产生的利息付清: 当月利息=上月剩余本金×月利率式2 其中月利率=年利率÷12。据传工商银行等某些银行在进行本金等额还款的计算方法中,月利率用了一个挺孙子的算法,这里暂且不提。 由上面利息偿还公式中可见,月利息是与上月剩余本金成正比的,由于在贷款初期,剩余本金较多,所以可见,贷款初期每月的利息较多,月还款额中偿还利息的份额较重。随着还款次数的增多,剩余本金将逐渐减少,月还款的利息也相应减少,直到最后一个月,本金全部还清,利息付最后一次,下个月将既无本金又无利息,至此,全部贷款偿还完毕。 两种贷款的偿还原理就如上所述。上述两个公式是月还款的基本公式,其他公式都可由此导出。下面我们就基于这两个公式推导一下两种还款方式的具体计算公式。 1. 等额本金还款方式 等额本金还款方式比较简单。顾名思义,这种方式下,每次还款的本金还款数是一样的。因此: 当月本金还款=总贷款数÷还款次数 当月利息=上月剩余本金×月利率 =总贷款数×(1-(还款月数-1)÷还款次数)×月利率

当月月还款额=当月本金还款+当月利息 =总贷款数×(1÷还款次数+(1-(还款月数-1)÷还款次数)×月利率) 总利息=所有利息之和 =总贷款数×月利率×(还款次数-(1+2+3+。。。+还款次数-1)÷还款次数) 其中1+2+3+…+还款次数-1是一个等差数列,其和为(1+还款次数-1)×(还款次数-1)/2=还款次数×(还款次数-1)/2 :总利息=总贷款数×月利率×(还款次数+1)÷2 由于等额本金还款每个月的本金还款额是固定的,而每月的利息是递减的,因此,等额本金还款每个月的还款额是不一样的。开始还得多,而后逐月递减。 等额本息还款方式: 等额本金还款,顾名思义就是每个月的还款额是固定的。由于还款利息是逐月减少的,因此反过来说,每月还款中的本金还款额是逐月增加的。 首先,我们先进行一番设定: 设:总贷款额=A 还款次数=B 还款月利率=C 月还款额=X 当月本金还款=Yn(n=还款月数) 先说第一个月,当月本金为全部贷款额=A,因此: 第一个月的利息=A×C 第一个月的本金还款额 Y1=X-第一个月的利息

房贷等额本息还款公式推导(详细)

等额本息还款公式推导 设贷款总额为A,银行月利率为β,总期数为m(个月),月还款额设为X,则各个月所欠银行贷款为: 第一个月A 第二个月A(1+β)-X 第三个月(A(1+β)-X)(1+β)-X=A(1+β)2-X[1+(1+β)]第四个月((A(1+β)-X)(1+β)-X)(1+β)-X =A(1+β)3-X[1+(1+β)+(1+β)2] … 由此可得第n个月后所欠银行贷款为 A(1+β)n –X[1+(1+β)+(1+β)2+…+(1+β)n-1]= A(1+β)n –X [(1+β)n-1]/β 由于还款总期数为m,也即第m月刚好还完银行所有贷款,因此有 A(1+β)m –X[(1+β)m-1]/β=0 由此求得

X = Aβ(1+β)m /[(1+β)m-1] ======================================================= ===== ◆关于A(1+β)n –X[1+(1+β)+(1+β)2+…+(1+β)n-1]= A(1+β)n –X[(1+β)n-1]/β的推导用了等比数列的求和公式 ◆1、(1+β)、(1+β)2、…、(1+β)n-1为等比数列 ◆关于等比数列的一些性质 (1)等比数列:An+1/An=q, n为自然数。 (2)通项公式:An=A1*q^(n-1); 推广式:An=Am·q^(n-m); (3)求和公式:Sn=nA1(q=1) Sn=[A1(1-q^n)]/(1-q) (4)性质: ①若m、n、p、q∈N,且m+n=p+q,则am·an=ap*aq; ②在等比数列中,依次每k项之和仍成等比数列. (5)“G是a、b的等比中项”“G^2=ab(G≠0)”. (6)在等比数列中,首项A1与公比q都不为零. ◆所以1+(1+β)+(1+β)2+…+(1+β)n-1 =[(1+β)n-1]/β 等额本金还款不同等额还款 问:等额本金还款是什么意思?与等额还款相比是否等额本金还款更省钱?

SIFT算法英文详解

SIFT: Scale Invariant Feature Transform The algorithm SIFT is quite an involved algorithm. It has a lot going on and can be come confusing, So I’ve split up the entire algorithm into multiple parts. Here’s an outline of what happens in SIFT. Constructing a scale space This is the initial preparation. You create internal representations of the original image to ensure scale invariance. This is done by generating a “scale space”. LoG Approximation The Laplacian of Gaussian is great for finding interesting points (or key points) in an image. But it’s computationally expensive. So we cheat and approximate it using the representation created earlier. Finding keypoints With the super fast approximation, we now try to find key points. These are maxima and minima in the Difference of Gaussian image we calculate in step 2 Get rid of bad key points Edges and low contrast regions are bad keypoints. Eliminating these makes the algorithm efficient and robust. A technique similar to the Harris Corner Detector is used here. Assigning an orientation to the keypoints An orientation is calculated for each key point. Any further calculations are done relative to this orientation. This effectively cancels out the effect of orientation, making it rotation invariant. Generate SIFT features Finally, with scale and rotation invariance in place, one more representation is generated. This helps uniquely identify features. Lets say you have 50,000 features. With this representation, you can easily identify the feature you’re looking for (sa y, a particular eye, or a sign board). That was an overview of the entire algorithm. Over the next few days, I’ll go through each step in detail. Finally, I’ll show you how to implement SIFT in OpenCV! What do I do with SIFT features? After you run through the algorithm, you’ll have SIFT features for your image. Once you have these, you can do whatever you want. Track images, detect and identify objects (which can be partly hidden as well), or whatever you can think of. We’ll get into this later as well. But the catch is, this algorithm is patented. >.< So, it’s good enough for academic purposes. But if you’re looking to make something commercial, look for something else! [Thanks to aLu for pointing out SURF is patented too] 1. Constructing a scale space Real world objects are meaningful only at a certain scale. You might see a sugar cube perfectly on a table. But if looking at the entire milky way, then it simply does not exist. This multi-scale nature of objects is quite common in nature. And a scale space attempts to replicate this concept

SIFT 特征提取算法详解

SIFT 特征提取算法总结 主要步骤 1)、尺度空间的生成; 2)、检测尺度空间极值点; 3)、精确定位极值点; 4)、为每个关键点指定方向参数; 5)、关键点描述子的生成。 L(x,y,σ), σ= 1.6 a good tradeoff

D(x,y,σ), σ= 1.6 a good tradeoff

关于尺度空间的理解说明:图中的2是必须的,尺度空间是连续的。在 Lowe 的论文中, 将第0层的初始尺度定为1.6,图片的初始尺度定为0.5. 在检测极值点前对原始图像的高斯平滑以致图像丢失高频信息,所以Lowe 建议在建立尺度空间前首先对原始图像长宽扩展一倍,以保留原始图像信息,增加特征点数量。尺度越大图像越模糊。 next octave 是由first octave 降采样得到(如2) , 尺度空间的所有取值,s为每组层数,一般为3~5 在DOG尺度空间下的极值点 同一组中的相邻尺度(由于k的取值关系,肯定是上下层)之间进行寻找

在极值比较的过程中,每一组图像的首末两层是无法进行极值比较的,为了满足尺度 变化的连续性,我们在每一组图像的顶层继续用高斯模糊生成了 3 幅图像, 高斯金字塔有每组S+3层图像。DOG金字塔每组有S+2层图像.

If ratio > (r+1)2/(r), throw it out (SIFT uses r=10) 表示DOG金字塔中某一尺度的图像x方向求导两次 通过拟和三维二次函数以精确确定关键点的位置和尺度(达到亚像素精度)?

直方图中的峰值就是主方向,其他的达到最大值80%的方向可作为辅助方向 Identify peak and assign orientation and sum of magnitude to key point The user may choose a threshold to exclude key points based on their assigned sum of magnitudes. 利用关键点邻域像素的梯度方向分布特性为每个关键点指定方向参数,使算子具备 旋转不变性。以关键点为中心的邻域窗口内采样,并用直方图统计邻域像素的梯度 方向。梯度直方图的范围是0~360度,其中每10度一个柱,总共36个柱。随着距中心点越远的领域其对直方图的贡献也响应减小.Lowe论文中还提到要使用高斯函 数对直方图进行平滑,减少突变的影响。

SIFT算法C语言逐步实现详解

SIFT算法C语言逐步实现详解(上) 引言: 在我写的关于sift算法的前倆篇文章里头,已经对sift算法有了初步的介绍:九、图像特征提取与匹配之SIFT算法,而后在:九(续)、sift算法的编译与实现里,我也简单记录下了如何利用opencv,gsl等库编译运行sift程序。 但据一朋友表示,是否能用c语言实现sift算法,同时,尽量不用到opencv,gsl等第三方库之类的东西。而且,Rob Hess维护的sift 库,也不好懂,有的人根本搞不懂是怎么一回事。 那么本文,就教你如何利用c语言一步一步实现sift算法,同时,你也就能真正明白sift算法到底是怎么一回事了。 ok,先看一下,本程序最终运行的效果图,sift 算法分为五个步骤(下文详述),对应以下第二--第六幅图:

sift算法的步骤 要实现一个算法,首先要完全理解这个算法的原理或思想。咱们先来简单了解下,什么叫sift算法: sift,尺度不变特征转换,是一种电脑视觉的算法用来侦测与描述影像中的局部性特征,它在空间尺度中寻找极值点,并提取出其位置、尺度、旋转不变量,此算法由David Lowe 在1999年所发表,2004年完善总结。 所谓,Sift算法就是用不同尺度(标准差)的高斯函数对图像进行平滑,然后比较平滑后图像的差别, 差别大的像素就是特征明显的点。 以下是sift算法的五个步骤: 一、建立图像尺度空间(或高斯金字塔),并检测极值点 首先建立尺度空间,要使得图像具有尺度空间不变形,就要建立尺度空间,sift算法采用了高斯函数来建立尺度空间,高斯函数公式为:

上述公式G(x,y,e),即为尺度可变高斯函数。 而,一个图像的尺度空间L(x,y,e) ,定义为原始图像I(x,y)与上述的一个可变尺度的2维高斯函数G(x,y,e) 卷积运算。 即,原始影像I(x,y)在不同的尺度e下,与高斯函数G(x,y,e)进行卷积,得到L(x,y,e),如下: 以上的(x,y)是空间坐标,e,是尺度坐标,或尺度空间因子,e的大小决定平滑程度,大尺度对应图像的概貌特征,小尺度对应图像的细节特征。大的e值对应粗糙尺度(低分辨率),反之,对应精细尺度(高分辨率)。 尺度,受e这个参数控制的表示。而不同的L(x,y,e)就构成了尺度空间,具体计算的时候,即使连续的高斯函数,都被离散为(一般为奇数大小)(2*k+1) *(2*k+1)矩阵,来和数字图像进行卷积运算。 随着e的变化,建立起不同的尺度空间,或称之为建立起图像的高斯金字塔。 但,像上述L(x,y,e) = G(x,y,e)*I(x,y)的操作,在进行高斯卷积时,整个图像就要遍历所有的像素进行卷积(边界点除外),于此,就造成了时间和空间上的很大浪费。 为了更有效的在尺度空间检测到稳定的关键点,也为了缩小时间和空间复杂度,对上述的操作作了一个改建:即,提出了高斯差分尺度空间(DOG scale-space)。利用不同尺度的高斯差分与原始图像I(x,y)相乘,卷积生成。 DOG算子计算简单,是尺度归一化的LOG算子的近似。 ok,耐心点,咱们再来总结一下上述内容: 1、高斯卷积 在组建一组尺度空间后,再组建下一组尺度空间,对上一组尺度空间的最后一幅图像进行二分之一采样,得到下一组尺度空间的第一幅图像,然后进行像建立第一组尺度空间那样的操作,得到第二组尺度空间,公式定义为 L(x,y,e) = G(x,y,e)*I(x,y)

等额本息法及等额本金法两种计算公式.doc

精品文档 等本息法和等本金法的两种算公式 一: 按等额本金还款 法:贷款额为: a, 月利率为: i , 年利率为: I , 还款月数: n, an 第 n 个月贷款剩余本金: a1=a, a2=a-a/n, a3=a-2*a/n ...次类推 还款利息总和为Y 每月应还本金: a/n 每月应还利息: an*i 每期还款 a/n +an*i 支付利息 Y=( n+1)*a*i/2 还款总额 =( n+1)*a*i/2+a 等本金法的算等本金(减法):算公式: 每月本金=款÷期数 第一个月的月供 =每月本金+款×月利率 第二个月的月供 =每月本金+(款-已本金)×月利率 申10 万 10 年个人住房商性款,算每月的月供款?(月利率: 4.7925 ‰)算果: 每月本金: 100000÷120= 833 元 第一个月的月供:833+ 100000×4.7925 ‰=1312.3 元 第二个月的月供:833+( 100000- 833)×4.7925 ‰= 1308.3 元 如此推?? 二 : 按等本息款法:款 a,月利率 i ,年利率 I ,款月数n,每月款 b,款利息和 Y 1: I =12×i 2: Y=n×b- a 3:第一月款利息:a×i 第二月款利息:〔a-( b- a×i )〕×i =( a×i -b)×( 1+ i ) ^1 +b 第三月款利息:{ a-( b- a×i )-〔 b-( a×i - b)×( 1+ i ) ^1 -b〕}×i =( a×i -b)×( 1+i ) ^2 + b 第四月款利息:=( a×i - b)×( 1+ i ) ^3 + b 第 n 月款利息:=(a×i - b)×( 1+ i ) ^( n- 1)+ b 求以上和:Y=( a×i -b)×〔( 1+ i ) ^n- 1〕÷i + n×b 4:以上两Y 相等求得 月均款 :b = a×i ×( 1+ i ) ^n ÷〔( 1+ i )^n - 1〕 支付利息 :Y = n×a×i ×( 1+i ) ^n ÷〔( 1+ i ) ^n - 1〕- a 款 :n ×a×i ×( 1+ i )^n ÷〔( 1+ i ) ^n- 1〕 注:a^b 表示 a 的 b 次方。 等本息法的算 ----- 例如下: 如款 21 万, 20 年,月利率 3.465 ‰按照上 面的等本息公式算 月均款 :b = a×i ×( 1+ i ) ^n ÷〔( 1+ i )^n - 1〕即: =1290.11017 即每个月款1290 元。 。 1欢迎下载

SIFT算法分析

SIFT算法分析 1 SIFT 主要思想 SIFT算法是一种提取局部特征的算法,在尺度空间寻找极值点,提取位置,尺度,旋转不变量。 2 SIFT 算法的主要特点: a)SIFT特征是图像的局部特征,其对旋转、尺度缩放、亮度变化保持不变性,对视角变化、仿射变换、噪声也保持一定程度的稳定性。 b)独特性(Distinctiveness)好,信息量丰富,适用于在海量特征数据库中进 行快速、准确的匹配。 c)多量性,即使少数的几个物体也可以产生大量SIFT特征向量。 d)高速性,经优化的SIFT匹配算法甚至可以达到实时的要求。 e)可扩展性,可以很方便的与其他形式的特征向量进行联合。 3 SIFT 算法流程图:

4 SIFT 算法详细 1)尺度空间的生成 尺度空间理论目的是模拟图像数据的多尺度特征。 高斯卷积核是实现尺度变换的唯一线性核,于是一副二维图像的尺度空间定义为: L( x, y, ) G( x, y, ) I (x, y) 其中G(x, y, ) 是尺度可变高斯函数,G( x, y, ) 2 1 2 y2 (x ) 2 e / 2 2 (x,y)是空间坐标,是尺度坐标。大小决定图像的平滑程度,大尺度对应图像的概貌特征,小尺度对应图像的细节特征。大的值对应粗糙尺度(低分辨率),反之,对应精细尺度(高分辨率)。 为了有效的在尺度空间检测到稳定的关键点,提出了高斯差分尺度空间(DOG scale-space)。利用不同尺度的高斯差分核与图像卷积生成。 D( x, y, ) (G( x, y,k ) G( x, y, )) I ( x, y) L( x, y,k ) L( x, y, ) DOG算子计算简单,是尺度归一化的LoG算子的近似。图像金字塔的构建:图像金字塔共O组,每组有S层,下一组的图像由上一 组图像降采样得到。 图1由两组高斯尺度空间图像示例金字塔的构建,第二组的第一副图像由第一组的第一副到最后一副图像由一个因子2降采样得到。图2 DoG算子的构建: 图1 Two octaves of a Gaussian scale-space image pyramid with s =2 intervals. The first image in the second octave is created by down sampling to last image in the previous

sift算法详解

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SIFT算法与RANSAC算法分析

概率论问题征解报告: (算法分析类) SIFT算法与RANSAC算法分析 班级:自23 姓名:黄青虬 学号:2012011438 作业号:146

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等额本息还款法

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遥感图像处理在汶川地震中的应用分析

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SIFT特征提取分析

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关于尺度空间的理解说明:2kσ中的2是必须的,尺度空间是连续的。在 Lowe的论文中,将第0层的初始尺度定为1.6(最模糊),图片的初始尺度定为0.5(最清晰). 在检测极值点前对原始图像的高斯平滑以致图像丢失高频信息,所以Lowe 建议在建立尺度空间前首先对原始图像长宽扩展一倍,以保留原始图像信息,增加特征点数量。尺度越大图像越模糊。 图像金字塔的建立:对于一幅图像I,建立其在不同尺度(scale)的图像,也成为子八度(octave),这是为了scale-invariant,也就是在任何尺度都能够有对应的特征点,第一个子八度的scale为原图大小,后面每个octave为上一个octave降采样的结果,即原图的1/4(长宽分别减半),构成下一个子八度(高一层金字塔)。

sift算法的MATLAB程序

% [image, descriptors, locs] = sift(imageFile) % % This function reads an image and returns its SIFT keypoints. % Input parameters: % imageFile: the file name for the image. % % Returned: % image: the image array in double format % descriptors: a K-by-128 matrix, where each row gives an invariant % descriptor for one of the K keypoints. The descriptor is a vector % of 128 values normalized to unit length. % locs: K-by-4 matrix, in which each row has the 4 values for a % keypoint location (row, column, scale, orientation). The % orientation is in the range [-PI, PI] radians. % % Credits: Thanks for initial version of this program to D. Alvaro and % J.J. Guerrero, Universidad de Zaragoza (modified by D. Lowe) function [image, descriptors, locs] = sift(imageFile) % Load image image = imread(imageFile); % If you have the Image Processing Toolbox, you can uncomment the following % lines to allow input of color images, which will be converted to grayscale. % if isrgb(image) % image = rgb2gray(image); % end [rows, cols] = size(image); % Convert into PGM imagefile, readable by "keypoints" executable f = fopen('tmp.pgm', 'w'); if f == -1 error('Could not create file tmp.pgm.'); end fprintf(f, 'P5\n%d\n%d\n255\n', cols, rows); fwrite(f, image', 'uint8'); fclose(f); % Call keypoints executable if isunix command = '!./sift '; else command = '!siftWin32 '; end command = [command ' tmp.key']; eval(command); % Open tmp.key and check its header g = fopen('tmp.key', 'r'); if g == -1

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